Optimal. Leaf size=26 \[ \sqrt {x^2-1} \log \left (\sqrt {x^2-1}+x\right )-x \]
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Rubi [A] time = 0.04, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {261, 2554, 8} \[ \sqrt {x^2-1} \log \left (\sqrt {x^2-1}+x\right )-x \]
Antiderivative was successfully verified.
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Rule 8
Rule 261
Rule 2554
Rubi steps
\begin {align*} \int \frac {x \log \left (x+\sqrt {-1+x^2}\right )}{\sqrt {-1+x^2}} \, dx &=\sqrt {-1+x^2} \log \left (x+\sqrt {-1+x^2}\right )-\int 1 \, dx\\ &=-x+\sqrt {-1+x^2} \log \left (x+\sqrt {-1+x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 26, normalized size = 1.00 \[ \sqrt {x^2-1} \log \left (\sqrt {x^2-1}+x\right )-x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 22, normalized size = 0.85 \[ \sqrt {x^{2} - 1} \log \left (x + \sqrt {x^{2} - 1}\right ) - x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.10, size = 22, normalized size = 0.85 \[ \sqrt {x^{2} - 1} \log \left (x + \sqrt {x^{2} - 1}\right ) - x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {x \ln \left (x +\sqrt {x^{2}-1}\right )}{\sqrt {x^{2}-1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 22, normalized size = 0.85 \[ \sqrt {x^{2} - 1} \log \left (x + \sqrt {x^{2} - 1}\right ) - x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {x\,\ln \left (x+\sqrt {x^2-1}\right )}{\sqrt {x^2-1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 9.12, size = 20, normalized size = 0.77 \[ - x + \sqrt {x^{2} - 1} \log {\left (x + \sqrt {x^{2} - 1} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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